Gilbreath Principle.pdf

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The Gilbreath Principle
JonathanBredin
September24,2001
1.Gilbreathprinciple
(a)Arrangecardssothatcolorsalternate
(b)Cutthedecksothatthebottomofeachhalfisadi‹erentcolor
(c)Ri � eshu � e
(d)Removethecardsinpairsfromthetop.Eachisofdi‹erentcolor.
(e)Why?
i.assumethatfirstcardtohittableisred
ii.ifnextcardcomesfromsameside,it’sblack
iii.othersideisalsoblack
iv.inanycase,thebottomsofthehalvesarestilldi‹erentcolors
2.moregeneralGilbreath
(a)orderdeckinsuits(spades,hearts,clubs,diamonds,spades,hearts,etc).
(b)dealcardstoformapile(reversingorder)
(c)ri � eshu � e
(d)drawquadruplets-willgetsetsofallfourcards
(e)canextendtotwoentiredecks
3.non-messing-uptheorem
(a)shu � e
(b)dealcardsfaceuptoformarectangle
(c)sorteachrowinincreasingorder
(d)sorteachcolumninincreasingorder
(e)noticethatrowsarestillordered
(f)why?
4.dilution
(a)diveddeckintoredandblackhalves
(b)take n cardsfromblackhalfandputintored
(c)shu � eredhalf
(d)take n cardsfromredhalfandputintoblack
(e)doestheredhalfhavemoreredcardsthantheblackhalfhasblackcards?
5.MontyHall
(a)guesswhichofthreecardshastheace
(b)revealwhichnon-guessedcarddoesnothaveanace
(c)whatisthebeststrategytofindtheacenow?
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(d)switchgivesp=2/3
6.Derivative
(a)dealthreecards
(b)twocardsofsamecolor
(c)whatisprobabilitythatthethirdcardisthatsamecolor?
(d)1/4
(e)suckerbet-giveevenodds
7.Parity
(a)takethreeredcardsfromthedeck
(b)putoneredoneback,takethreeblackcards
(c)putoneblackoneback,takethreered
(d)repeat
(e)cantherebethesamenumberofredandblackcards?
(f)no,parity
(g)alwaysholdinganoddnumberofcards
8.Pickfour
(a)placecardsfacedownace-9inincreasingorder(aceleft)
(b)removeacardfromeitherend
(c)removeacardfromeitherend
(d)removeacardfromeitherend
(e)addvalueofthreecardsanddivideby6= n
(f)turnoverthe n thcardleft-right
(g)it’sthefour
9.findunknowncard
(a)divide21cardsinto3equalgroups
(b)observerchooses1cardfromagroup,
thecardissecretbutthepileisnot
(c)placethechosenpilebetweentheothertwo
(d)dealout3pilesofsevenagain
(e)askwhichpilethechosencardisin
(f)placechosenpileinbetweenothers
(g)dealagain
(h)identifypile
(i)placechosenpilebetweenothertwo
(j)secretcardisthe11thatthetopofthedeck
(k)why?
i.ifwepickarandomcardinapile,sayitisthe a thintheorderofthepile
ii.thenweplaceitinthe7+ a -thslotwhenwestacktheothertwopiles
iii.thenthecardisinthe b (7+ a ) = 3 c +1positioninthenewpile....
iv.attheendthecardisintheposition
b b 7+ a
3
c +1+7
3
c +1
2
 
v.thepossiblevaluesfortheinnerfloorare2,3,or4
vi.thismeansthatthepossiblevaluesforthenextfloorvalueare3(thewholepartof10,11,12divided
by3)
vii.addonesoourchosencardisfourthinitsstack
viii.thereare7cardsaheadofourcardoncewestackthepiles
ix.sothechosencardisthe11th
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